13 research outputs found
Discrete maximal regularity of time-stepping schemes for fractional evolution equations
In this work, we establish the maximal -regularity for several time
stepping schemes for a fractional evolution model, which involves a fractional
derivative of order , , in time. These schemes
include convolution quadratures generated by backward Euler method and
second-order backward difference formula, the L1 scheme, explicit Euler method
and a fractional variant of the Crank-Nicolson method. The main tools for the
analysis include operator-valued Fourier multiplier theorem due to Weis [48]
and its discrete analogue due to Blunck [10]. These results generalize the
corresponding results for parabolic problems
Discrete maximal regularity for the finite element approximation of the Stokes operator and its application
Maximal regularity for the Stokes operator plays a crucial role in the theory
of the non-stationary Navier--Stokes equations. In this paper, we consider the
finite element semi-discretization of the non-stationary Stokes problem and
establish the discrete counterpart of maximal regularity in for . For the proof of discrete
maximal regularity, we introduce the temporally regularized Green's function.
With the aid of this notion, we prove discrete maximal regularity without the
Gaussian estimate. As an application, we present -type
error estimates for the approximation of the non-stationary Stokes problem